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Enumeration of Intersection Graphs of x-Monotone Curves

Authors Jacob Fox, János Pach, Andrew Suk

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ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Enumeration
  • intersection graphs
  • pseudo-segments
  • x-monotone

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Abstract

A curve in the plane is x-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct 2^Ω(n^{4/3}) families, each consisting of n labelled x-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most 2^O(n^{3/2-ε}), where ε > 0 is a suitable constant. Our proof uses an upper bound on the number of set systems of size m on a ground set of size n, with VC-dimension at most d. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

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Jacob Fox, János Pach, and Andrew Suk. Enumeration of Intersection Graphs of x-Monotone Curves. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.4

Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, CA, USA
János Pach
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

Funding

  • Fox, Jacob: supported by NSF Award DMS-2154129.
  • Pach, János: Rényi Institute, Budapest. Supported by NKFIH grant K-176529, Austrian Science Fund Z 342-N31 and ERC Advanced Grant "GeoScape."
  • Suk, Andrew: Supported by NSF CAREER award DMS-1800746, and NSF awards DMS-1952786 and DMS-2246847.

Acknowledgements

We would like to thank Zixiang Xu for pointing out the reference [N. Alon et al., 2017] to us.

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